This is a multi-part message in MIME format. ---------------0604281027472 Content-Type: text/plain; name="06-137.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-137.comments" 25 pages ---------------0604281027472 Content-Type: text/plain; name="06-137.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-137.keywords" ising model; long range interactions; periodic ground states. ---------------0604281027472 Content-Type: text/plain; name="ising.bib" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="ising.bib" @book{B01, author = "P. Ball", title = "The self--made tapestry: pattern formation in nature", publisher = "Oxford Univ. Press", year = "2001", } @article{AWMD95, author = "J. Arlett and J. P. Whitehead and A. B. MacIsaac and K. De'Bell", title = "Phase diagram for the striped phase in the two-dimensional dipolar Ising model", journal = "Phys. Rev. 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Phys.", volume = "79", pages = "25--32", year = "1981", } ---------------0604281027472 Content-Type: application/x-tex; name="ising.bbl" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ising.bbl" \begin{thebibliography}{34} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname bibnamefont\endcsname\relax \def\bibnamefont#1{#1}\fi \expandafter\ifx\csname bibfnamefont\endcsname\relax \def\bibfnamefont#1{#1}\fi \expandafter\ifx\csname citenamefont\endcsname\relax \def\citenamefont#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{\texttt{#1}}\fi \expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi \providecommand{\bibinfo}[2]{#2} \providecommand{\eprint}[2][]{\url{#2}} \bibitem[{\citenamefont{Ball}(2001)}]{B01} \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Ball}}, \emph{\bibinfo{title}{The self--made tapestry: pattern formation in nature}} (\bibinfo{publisher}{Oxford Univ. Press}, \bibinfo{year}{2001}). \bibitem[{\citenamefont{Muratov}(2002)}]{M02} \bibinfo{author}{\bibfnamefont{C.~B.} \bibnamefont{Muratov}}, \bibinfo{journal}{Physical Review E} \textbf{\bibinfo{volume}{66}}, \bibinfo{pages}{066108} (\bibinfo{year}{2002}). \bibitem[{\citenamefont{Bowman and Newell}(1998)}]{BN98} \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Bowman}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{A.~C.} \bibnamefont{Newell}}, \bibinfo{journal}{Rev. Mod. Phys.} \textbf{\bibinfo{volume}{70}}, \bibinfo{pages}{289} (\bibinfo{year}{1998}). \bibitem[{\citenamefont{Malescio and Pellicane}(2003)}]{MP03} \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Malescio}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Pellicane}}, \bibinfo{journal}{Nature Materials} \textbf{\bibinfo{volume}{2}}, \bibinfo{pages}{97} (\bibinfo{year}{2003}). \bibitem[{\citenamefont{Seul and Andelman}(1995)}]{SA95} \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Seul}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Andelman}}, \bibinfo{journal}{Science} \textbf{\bibinfo{volume}{267}}, \bibinfo{pages}{476} (\bibinfo{year}{1995}). \bibitem[{\citenamefont{Stoycheva and Singer}(1999)}]{SS99} \bibinfo{author}{\bibfnamefont{A.~D.} \bibnamefont{Stoycheva}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.~J.} \bibnamefont{Singer}}, \bibinfo{journal}{Physical Review Letters} \textbf{\bibinfo{volume}{84}}, \bibinfo{pages}{4657} (\bibinfo{year}{1999}). \bibitem[{\citenamefont{Seul and Wolfe}(1992)}]{SW92} \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Seul}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Wolfe}}, \bibinfo{journal}{Phys. Rev. A} \textbf{\bibinfo{volume}{46}}, \bibinfo{pages}{7519} (\bibinfo{year}{1992}). \bibitem[{\citenamefont{Mohwald}(1988)}]{M88} \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Mohwald}}, \bibinfo{journal}{Thin Solid Films} \textbf{\bibinfo{volume}{159}}, \bibinfo{pages}{1} (\bibinfo{year}{1988}). \bibitem[{\citenamefont{Keller and McConnell}(1999)}]{KM99} \bibinfo{author}{\bibfnamefont{S.~L.} \bibnamefont{Keller}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{H.~M.} \bibnamefont{McConnell}}, \bibinfo{journal}{Physical Review Letters} \textbf{\bibinfo{volume}{82}}, \bibinfo{pages}{1602} (\bibinfo{year}{1999}). \bibitem[{\citenamefont{Maclennan and Seul}(1992)}]{MS92} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Maclennan}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Seul}}, \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{69}}, \bibinfo{pages}{2082} (\bibinfo{year}{1992}). \bibitem[{\citenamefont{Harrison et~al.}(2000)\citenamefont{Harrison, Adamson, Cheng, Sebastian, Sethuraman, Huse, Register, and Chaikin}}]{H00} \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Harrison}}, \bibinfo{author}{\bibfnamefont{D.~H.} \bibnamefont{Adamson}}, \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Cheng}}, \bibinfo{author}{\bibfnamefont{J.~M.} \bibnamefont{Sebastian}}, \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Sethuraman}}, \bibinfo{author}{\bibfnamefont{D.~A.} \bibnamefont{Huse}}, \bibinfo{author}{\bibfnamefont{R.~A.} \bibnamefont{Register}}, \bibnamefont{and} \bibinfo{author}{\bibfnamefont{P.~M.} \bibnamefont{Chaikin}}, \bibinfo{journal}{Science} \textbf{\bibinfo{volume}{290}}, \bibinfo{pages}{1558} (\bibinfo{year}{2000}). \bibitem[{\citenamefont{MacIsaac et~al.}(1995)\citenamefont{MacIsaac, Whitehead, Robinson, and De'Bell}}]{MWRD95} \bibinfo{author}{\bibfnamefont{A.~B.} \bibnamefont{MacIsaac}}, \bibinfo{author}{\bibfnamefont{J.~P.} \bibnamefont{Whitehead}}, \bibinfo{author}{\bibfnamefont{M.~C.} \bibnamefont{Robinson}}, \bibnamefont{and} \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{De'Bell}}, \bibinfo{journal}{Physical Review B} \textbf{\bibinfo{volume}{51}}, \bibinfo{pages}{16033} (\bibinfo{year}{1995}). \bibitem[{\citenamefont{Arlett et~al.}(1996)\citenamefont{Arlett, Whitehead, MacIsaac, and De'Bell}}]{AWMD95} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Arlett}}, \bibinfo{author}{\bibfnamefont{J.~P.} \bibnamefont{Whitehead}}, \bibinfo{author}{\bibfnamefont{A.~B.} \bibnamefont{MacIsaac}}, \bibnamefont{and} \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{De'Bell}}, \bibinfo{journal}{Phys. 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Phys.} \textbf{\bibinfo{volume}{40}}, \bibinfo{pages}{4956} (\bibinfo{year}{1999}). \bibitem[{\citenamefont{Ginibre et~al.}(1966)\citenamefont{Ginibre, Grossmann, and Ruelle}}]{GGR66} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Ginibre}}, \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Grossmann}}, \bibnamefont{and} \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Ruelle}}, \bibinfo{journal}{Communications in Mathematical Physics} \textbf{\bibinfo{volume}{3}}, \bibinfo{pages}{187} (\bibinfo{year}{1966}). \bibitem[{\citenamefont{Ruelle}(1968)}]{R68} \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Ruelle}}, \bibinfo{journal}{Communications in Mathematical Physics} \textbf{\bibinfo{volume}{9}}, \bibinfo{pages}{267} (\bibinfo{year}{1968}). \bibitem[{\citenamefont{van Enter}(1981)}]{VE81} \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{van Enter}}, \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{79}}, \bibinfo{pages}{25} (\bibinfo{year}{1981}). \bibitem[{\citenamefont{Frohlich et~al.}(1980)\citenamefont{Frohlich, Israel, Lieb, and Simon}}]{FILS80} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Frohlich}}, \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Israel}}, \bibinfo{author}{\bibfnamefont{E.}~\bibnamefont{Lieb}}, \bibnamefont{and} \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Simon}}, \bibinfo{journal}{Journal of Statistical Physics} \textbf{\bibinfo{volume}{22}}, \bibinfo{pages}{297} (\bibinfo{year}{1980}). \bibitem[{\citenamefont{Bak and Bruinsma}(1982)}]{BB82} \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Bak}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Bruinsma}}, \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{49}}, \bibinfo{pages}{249} (\bibinfo{year}{1982}). \bibitem[{\citenamefont{Dyson}(1969)}]{D69} \bibinfo{author}{\bibfnamefont{F.~J.} \bibnamefont{Dyson}}, \bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{12}}, \bibinfo{pages}{91} (\bibinfo{year}{1969}). \bibitem[{\citenamefont{Frohlich and Spencer}(1982)}]{FS82} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Frohlich}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Spencer}}, \bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{84}}, \bibinfo{pages}{87} (\bibinfo{year}{1982}). \bibitem[{\citenamefont{Fannes et~al.}(1982)\citenamefont{Fannes, Vanheuverzwijn, and Verbeure}}]{FVV82} \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Fannes}}, \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Vanheuverzwijn}}, \bibnamefont{and} \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Verbeure}}, \bibinfo{journal}{Jour. Stat. Phys.} \textbf{\bibinfo{volume}{29}}, \bibinfo{pages}{547} (\bibinfo{year}{1982}). \bibitem[{\citenamefont{Burkov and Sinai}(1983)}]{BS83} \bibinfo{author}{\bibfnamefont{S.~E.} \bibnamefont{Burkov}} \bibnamefont{and} \bibinfo{author}{\bibfnamefont{Y.~G.} \bibnamefont{Sinai}}, \bibinfo{journal}{Uspekhi Mat. Nauk} \textbf{\bibinfo{volume}{38}}, \bibinfo{pages}{205} (\bibinfo{year}{1983}). \bibitem[{\citenamefont{Bricmont et~al.}(1979)\citenamefont{Bricmont, Lebowitz, and Pfister}}]{BLP79} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Bricmont}}, \bibinfo{author}{\bibfnamefont{J.~L.} \bibnamefont{Lebowitz}}, \bibnamefont{and} \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Pfister}}, \bibinfo{journal}{Jour. Stat. Phys.} \textbf{\bibinfo{volume}{21}}, \bibinfo{pages}{573} (\bibinfo{year}{1979}). \bibitem[{\citenamefont{Khanin}()}]{Kh} \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Khanin}}, \bibinfo{note}{private communication}. \end{thebibliography} ---------------0604281027472 Content-Type: application/x-tex; name="ising.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ising.tex" \documentclass[11pt]{revtex4} \setlength{\oddsidemargin}{0cm} \setlength{\evensidemargin}{0cm} \setlength{\textwidth}{14.5cm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MACRO%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcount\driver \newcount\bozza %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% DEFINIZIONI DI FONT %%%%%%%%%%%%%%%%%%%%%%%%%%% 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scaled\magstep1 %\font\msytwwww=msbm4 scaled\magstep1 \font\indbf=cmbx10 scaled\magstep2 \font\type=cmtt10 \def\st{\scriptstyle} \font\ottorm=cmr8\font\ottoi=cmmi8\font\ottosy=cmsy8% \font\ottobf=cmbx8\font\ottott=cmtt8% \font\ottocss=cmcsc8% \font\ottosl=cmsl8\font\ottoit=cmti8% \font\sixrm=cmr6\font\sixbf=cmbx6\font\sixi=cmmi6\font\sixsy=cmsy6% \font\fiverm=cmr5\font\fivesy=cmsy5 \font\fivei=cmmi5 \font\fivebf=cmbx5% \def\ottopunti{\def\rm{\fam0\ottorm}% \textfont0=\ottorm\scriptfont0=\sixrm\scriptscriptfont0=\fiverm% \textfont1=\ottoi\scriptfont1=\sixi\scriptscriptfont1=\fivei% \textfont2=\ottosy\scriptfont2=\sixsy\scriptscriptfont2=\fivesy% %\textfont3=\tenex\scriptfont3=\tenex\scriptscriptfont3=\tenex% \textfont4=\ottocss\scriptfont4=\sc\scriptscriptfont4=\sc% \scriptfont4=\ottocss\scriptscriptfont4=\ottocss% \textfont5=\tenmib\scriptfont5=\sevenmib\scriptscriptfont5=\fivei %\textfont\itfam=\ottoit\def\it{\fam\itfam\ottoit}% %\textfont\slfam=\ottosl\def\sl{\fam\slfam\ottosl}% %\textfont\ttfam=\ottott\def\tt{\fam\ttfam\ottott}% %\textfont\bffam=\ottobf\scriptfont\bffam=\sixbf% %\scriptscriptfont\bffam=\fivebf\def\bf{\fam\bffam\ottobf}% %\tt\ttglue=.5em plus.25em minus.15em% \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \normalbaselineskip=9pt\let\sc=\sixrm\normalbaselines\rm} \let\nota=\ottopunti% \mathchardef\BDpr = "0540 %Dpr \mathchardef\Bg = "050D %gamma \def\sf{\textfont1=\amit} {\count255=\time\divide\count255 by 60 \xdef\hourmin{\number\count255} \multiply\count255 by-60\advance\count255 by\time \xdef\hourmin{\hourmin:\ifnum\count255<10 0\fi\the\count255}} %\def\openone{\leavevmode\hbox{\ninerm 1\kern-3.3pt\tenrm1}}% %\def\*{\vglue0.3truecm} %\ifnum\mgnf=1 \def\openone{\leavevmode\hbox{\elevenrm 1\kern-3.63pt\twelverm1}}% \def\*{\vglue0.5truecm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% SIMBOLI VARI %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \let\z=\zeta \let\h=\eta \let\th=\theta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho \let\s=\sigma \let\t=\tau \let\f=\varphi \let\ph=\varphi\let\c=\chi \let\ps=\psi \let\y=\upsilon \let\o=\omega\let\si=\varsigma \let\G=\Gamma \let\D=\Delta \let\Th=\Theta\let\L=\Lambda \let\X=\Xi \let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega \let\Y=\Upsilon \def\\{\hfill\break} \let\==\equiv \let\txt=\textstyle\let\dis=\displaystyle \let\io=\infty \def\Dpr{\V\dpr\,} \def\aps{{\it a posteriori\ }}\def\ap{{\it a priori\ }} \let\0=\noindent\def\pagina{{\vfill\eject}} \def\bra#1{{\langle#1|}}\def\ket#1{{|#1\rangle}} \def\media#1{{\langle#1\rangle}} %\def\ie{\hbox{\it i.e.\ }}\def\eg{\hbox{\it e.g.\ }} \def\ie{{i.e. }}\def\eg{{e.g. }} \let\dpr=\partial \def\der{{\rm d}} \let\circa=\cong \def\arccot{{\rm arccot}} \def\tende#1{\,\vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}\,} \def\circage{\lower2pt\hbox{$\,\buildrel > \over {\scriptstyle \sim}\,$}} \def\otto{\,{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\,} \def\fra#1#2{{#1\over#2}} \def\PPP{{\cal P}}\def\EE{{\cal E}}\def\MM{{\cal M}} \def\VV{{\cal V}} \def\CC{{\cal C}}\def\FF{{\cal F}} \def\HHH{{\cal H}}\def\WW{{\cal W}} \def\TT{{\cal T}}\def\NN{{\cal N}} \def\BBB{{\cal B}}\def\III{{\cal I}} \def\RR{{\cal R}}\def\LL{{\cal L}} \def\JJ{{\cal J}} \def\OO{{\cal O}} \def\DD{{\cal D}}\def\AAA{{\cal A}}\def\GG{{\cal G}} \def\SS{{\cal S}} \def\KK{{\cal K}}\def\UU{{\cal U}} \def\QQ{{\cal Q}} \def\XXX{{\cal X}} \def\T#1{{#1_{\kern-3pt\lower7pt\hbox{$\widetilde{}$}}\kern3pt}} \def\VVV#1{{\underline #1}_{\kern-3pt \lower7pt\hbox{$\widetilde{}$}}\kern3pt\,} \def\W#1{#1_{\kern-3pt\lower7.5pt\hbox{$\widetilde{}$}}\kern2pt\,} \def\Re{{\rm Re}\,}\def\Im{{\rm Im}\,} \def\lis{\overline}\def\tto{\Rightarrow} \def\etc{{\it etc}} \def\acapo{\hfill\break} \def\mod{{\rm mod}\,} \def\per{{\rm per}\,} \def\sign{{\rm sign}\,} \def\indica{\leaders \hbox to 0.5cm{\hss.\hss}\hfill} \def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill} \def\hh{{\bf h}} \def\HH{{\bf H}} \def\AA{{\bf A}} \def\qq{{\bf q}} \def\BB{{\bf B}} \def\XX{{\bf X}} \def\PP{{\bf P}} \def\pp{{\bf p}} \def\vv{{\bf v}} \def\xx{{\bf x}} \def\yy{{\bf y}} \def\zz{{\bf z}} \def\aaa{{\bf a}}\def\bbb{{\bf b}}\def\hhh{{\bf h}}\def\II{{\bf I}} \def\ul{\underline} \def\olu{{\overline{u}}} \def\defin{{\buildrel def\over=}} \def\wt{\widetilde} \def\wh{\widehat} \def\Dpr{\BDpr\,} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% LETTERE GRECHE E LATINE IN NERETTO %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % lettere greche e latine in neretto italico - pag.430 del manuale \mathchardef\aa = "050B \mathchardef\bb = "050C \mathchardef\ggg = "050D \mathchardef\xxx = "0518 %\mathchardef\hhh = "0511 \mathchardef\zzzzz= "0510 \mathchardef\oo = "0521 \mathchardef\lll = "0515 \mathchardef\mm = "0516 \mathchardef\Dp = "0540 \mathchardef\H = "0548 \mathchardef\FFF = "0546 \mathchardef\ppp = "0570 \mathchardef\Bn = "0517 %\mathchardef\ff = "0527 \mathchardef\pps = "0520 %\mathchardef\XXX = "0504 \mathchardef\fff = "0527 \mathchardef\FFF = "0508 \mathchardef\nnnnn= "056E \def\ol#1{{\overline #1}} \def\to{\rightarrow} \def\la{\left\langle} \def\ra{\right\rangle} \def\Overline#1{{\bar#1}} \let\ciao=\bye \def\qed{\hfill\raise1pt\hbox{\vrule height5pt width5pt depth0pt}} \def\hf#1{{\hat \f_{#1}}} \def\barf#1{{\tilde \f_{#1}}} \def\tg#1{{\tilde g_{#1}}} \def\bq{{\bar q}} %\def\Val{{\bf Val}} \def\Val{{\rm Val}} \def\indic{\hbox{\raise-2pt \hbox{\indbf 1}}} \def\RRR{\hbox{\msytw R}} \def\rrrr{\hbox{\msytww R}} \def\rrr{\hbox{\msytwww R}} \def\CCC{\hbox{\msytw C}} \def\cccc{\hbox{\msytww C}} \def\ccc{\hbox{\msytwww C}} \def\NNN{\hbox{\msytw N}} \def\nnnn{\hbox{\msytww N}} \def\nnn{\hbox{\msytwww N}} \def\ZZZ{\hbox{\msytw Z}} \def\zzzz{\hbox{\msytww Z}} \def\zzz{\hbox{\msytwww Z}} \def\SSS{\hbox{\msytw S}} \def\ssss{\hbox{\msytww S}} \def\sss{\hbox{\msytwww S}} \def\TTT{\hbox{\msytw T}} \def\tttt{\hbox{\msytww T}} \def\ttt{\hbox{\msytwww T}}\def\MMM{\hbox{\euftw M}} \def\QQQ{\hbox{\msytw Q}} \def\qqqq{\hbox{\msytww Q}} \def\qqq{\hbox{\msytwww Q}} \def\vvv{\hbox{\euftw v}} \def\vvvv{\hbox{\euftww v}} \def\vvvvv{\hbox{\euftwww v}}\def\www{\hbox{\euftw w}} \def\wwww{\hbox{\euftww w}} \def\wwwww{\hbox{\euftwww w}} \def\vvr{\hbox{\euftw r}} \def\vvvr{\hbox{\euftww r}} \def\ul#1{{\underline#1}} \def\Sqrt#1{{\sqrt{#1}}} %\def\Sqrt#1{{{#1}^{\fra{1}{2}}}} \def\V0{{\bf 0}} \def\defi{\,{\buildrel def\over=}\,} \def\lhs{{\it l.h.s.}\ } \def\rhs{{\it r.h.s.}\ } % lettere greche e latine in neretto italico - pag.430 del manuale \font\tenmib=cmmib10 \font\eightmib=cmmib8 \font\sevenmib=cmmib7\font\fivemib=cmmib5 \font\ottoit=cmti8 \font\fiveit=cmti5\font\sixit=cmti6%% %!!!@@@\font\fiveit=cmti7\font\sixit=cmti7%% \font\fivei=cmmi5\font\sixi=cmmi6\font\ottoi=cmmi8 \font\ottorm=cmr8\font\fiverm=cmr5\font\sixrm=cmr6 \font\ottosy=cmsy8\font\sixsy=cmsy6\font\fivesy=cmsy5%% \font\ottobf=cmbx8\font\sixbf=cmbx6\font\fivebf=cmbx5% \font\ottott=cmtt8% \font\ottocss=cmcsc8% \font\ottosl=cmsl8% \textfont5=\tenmib\scriptfont5=\sevenmib\scriptscriptfont5=\fivemib \mathchardef\Ba = "050B %alfa \mathchardef\Bb = "050C %beta \mathchardef\Bg = "050D %gamma \mathchardef\Bd = "050E %delta \mathchardef\Be = "0522 %varepsilon \mathchardef\Bee = "050F %epsilon \mathchardef\Bz = "0510 %zeta \mathchardef\Bh = "0511 %eta \mathchardef\Bthh = "0512 %teta \mathchardef\Bth = "0523 %varteta \mathchardef\Bi = "0513 %iota \mathchardef\Bk = "0514 %kappa \mathchardef\Bl = "0515 %lambda \mathchardef\Bm = "0516 %mu \mathchardef\Bn = "0517 %nu \mathchardef\Bx = "0518 %xi \mathchardef\Bom = "0530 %omi \mathchardef\Bp = "0519 %pi \mathchardef\Br = "0525 %ro \mathchardef\Bro = "051A %varrho \mathchardef\Bs = "051B %sigma \mathchardef\Bsi = "0526 %varsigma \mathchardef\Bt = "051C %tau \mathchardef\Bu = "051D %upsilon \mathchardef\Bf = "0527 %phi \mathchardef\Bff = "051E %varphi \mathchardef\Bch = "051F %chi \mathchardef\Bps = "0520 %psi \mathchardef\Bo = "0521 %omega \mathchardef\Bome = "0524 %varomega \mathchardef\BG = "0500 %Gamma \mathchardef\BD = "0501 %Delta \mathchardef\BTh = "0502 %Theta \mathchardef\BL = "0503 %Lambda \mathchardef\BX = "0504 %Xi \mathchardef\BP = "0505 %Pi \mathchardef\BS = "0506 %Sigma \mathchardef\BU = "0507 %Upsilon \mathchardef\BF = "0508 %Fi \mathchardef\BPs = "0509 %Psi \mathchardef\BO = "050A %Omega \mathchardef\BDpr = "0540 %Dpr \mathchardef\Bstl = "053F %* \def\BK{\bf K} \def\V#1{{\bf#1}} \let\aa=\Ba\let\fff=\Bf\let\defin=\defi\def\HHH{{\cal H}} \let\wt=\widetilde\def\AAA{{\cal A}}\let\oo=\Bo\let\nn=\Bn \let\aaa=\Ba\let\pps=\Bps\def\hhh={\V h}\def\bbb{{\V b}} \let\bb=\Bb\def\ss{\ul{\s}} \def\RRR{\hbox{\msytw R}} \def\rrrr{\hbox{\msytww R}} \def\rrr{\hbox{\msytwww R}} \def\CCC{\hbox{\msytw C}} \def\cccc{\hbox{\msytww C}} \def\ccc{\hbox{\msytwww C}} \def\NNN{\hbox{\msytw N}} \def\nnnn{\hbox{\msytww N}} \def\nnn{\hbox{\msytwww N}} \def\ZZZ{\hbox{\msytw Z}} \def\zzzz{\hbox{\msytww Z}} \def\zzz{\hbox{\msytwww Z}} \def\TTT{\hbox{\msytw T}} \def\tttt{\hbox{\msytww T}} \def\ttt{\hbox{\msytwww T}} \def\QQQ{\hbox{\msytw Q}} \def\qqqq{\hbox{\msytww Q}} \def\qqq{\hbox{\msytwww Q}} %\def\vvvv{\hbox{\msytww V}}\def\vvvvv{\hbox{\msytww V}} %\def\wwww{\hbox{\msytww W}}\def\wwwww{\hbox{\msytww W}} \let\ul=\underline\def\hh{{\V h}} \def\cfr{{cf. }}\let\ig=\int \def\Tr{{\rm Tr}} \def\dist{{\rm dist}} %%% INSERIMENTO FIGURE % % Se si usa DVIPS e si vuole utilizzare delle macro %postscript personali, contenute % nel file ini.ps, togliere il commento alla riga seguente %\ifnum\driver=1 \special{header=ini.pst} \fi % % Il comando seguente inserisce una scatola contenente #3 in modo che % l'angolo superiore sinistro occupi la posizione (#1,#2) % \def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} % % Il comando seguente crea una scatola di dimensioni #1x#2 contenente % il disegno descritto in #4.ps o in #4.pdf; % in questo disegno si possono introdurre delle stringhe usando \ins % e mettendo le istruzioni relative nell'argomento #3. % Il file #4.ps contiene le istruzioni postscript, che devono essere scritte % presupponendo che l'origine sia nell'angolo inferiore sinistro della % scatola, mentre per il resto l'ambiente grafico e' quello standard. % Il file #4.pdf contiene una figura in formato pdf, che viene posizionata % nello stesso modo. % #5 deve essere della forma \eqg("nome simbolico"). % % Le istruzioni postscript possono essere inserite nel file che contiene % l'istruzione \insertplot, racchiudendole fra le istruzioni \initfig{#4} % e \endfig; inoltre ogni riga deve cominciare con "write13<" e deve finire % con ">". In questo modo si crea il file #4.ps relativo alla figura. % \newdimen\xshift \newdimen\xwidth \newdimen\yshift \newcount\griglia \def\insertplot#1#2#3#4#5#6{% \begin{figure}[h] \begin{center} \vspace{#2pt} \begin{minipage}{#1pt} #3 \ifnum\driver=1 \griglia=#6 \ifnum\griglia=1 \openout13=griglia.ps \write13{gsave .2 setlinewidth} \write13{0 10 #1 {dup 0 moveto #2 lineto } for} \write13{0 10 #2 {dup 0 exch moveto #1 exch lineto } for} \write13{stroke} \write13{.5 setlinewidth} \write13{0 50 #1 {dup 0 moveto #2 lineto } for} \write13{0 50 #2 {dup 0 exch moveto #1 exch lineto } for} \write13{stroke grestore} \closeout13 \special{psfile=griglia.ps}\fi \special{psfile=#4.ps}\fi \ifnum\driver=2 \special{pdf:epdf (#4.pdf)}\fi \end{minipage} \end{center} \caption{#5} \end{figure} } %\def\ltopg{\,{\raise-1ex\hbox{$\sf <$}\atop \raise1ex\hbox{$\sf >$}}\,} \def\gtopl{\hbox{\msxtw \char63}} \def\ltopg{\hbox{\msxtw \char55}} \newdimen\shift \shift=-1truecm \def\lb#1{% \ifnum\bozza=1 \label{#1}\rlap{\kern\shift{$\scriptstyle#1$}} \else\label{#1} \fi} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bea{\begin{eqnarray}}\def\eea{\end{eqnarray}} \def\bean{\begin{eqnarray*}}\def\eean{\end{eqnarray*}} \def\bfr{\begin{flushright}}\def\efr{\end{flushright}} \def\bc{\begin{center}}\def\ec{\end{center}} \def\ba#1{\begin{array}{#1}} \def\ea{\end{array}} \def\bd{\begin{description}}\def\ed{\end{description}} \def\bv{\begin{verbatim}}\def\ev{\end{verbatim}} \def\nn{\nonumber} \def\Halmos{\hfill\vrule height10pt width4pt depth2pt \par\hbox to \hsize{}} \def\pref#1{(\ref{#1})} \def\Dim{{\bf Dim. -\ \ }} \def\Sol{{\bf Soluzione -\ \ }} \def\virg{\quad,\quad} \def\bsl{$\backslash$} \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}{Theorem}[section] \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \newdimen\xshift \newdimen\xwidth \newdimen\yshift \newdimen\ywidth \def\ins#1#2#3{\vbox to0pt{\kern-#2\hbox{\kern#1 #3}\vss}\nointerlineskip} \def\eqfig#1#2#3#4#5{ \par\xwidth=#1 \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2 \yshift=#2 \divide\yshift by 2 \line{\hglue\xshift \vbox to #2{\vfil #3 \special{psfile=#4.ps} }\hfill\raise\yshift\hbox{#5}}} \def\8{\write12} \def\figini#1{ \catcode`\%=12\catcode`\{=12\catcode`\}=12 \catcode`\<=1\catcode`\>=2 \openout12=#1.ps} \def\figfin{ \closeout12 \catcode`\%=14\catcode`\{=1% \catcode`\}=2\catcode`\<=12\catcode`\>=12} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \driver=1 \bozza=0 \font\bigtenrm=cmr10 scaled \magstep1 \font\bigteni=cmmi10 scaled \magstep1 \begin{document} \title{Ising models with long--range dipolar\\ and short range ferromagnetic interactions} \* \author{Alessandro Giuliani} \affiliation{Department of Physics, Princeton University, Princeton 08544 NJ, USA} \author{Joel L. Lebowitz} \affiliation{Department of Mathematics and Physics, Rutgers University, Piscataway, NJ 08854 USA.} \author{Elliott H. Lieb} \affiliation{Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08544 USA.} \vspace{1cm} \date{\today} \begin{abstract} We study the ground state of a $d$--dimensional Ising model with both long range (dipole--like) and nearest neighbor ferromagnetic (FM) interactions. The long range interaction is equal to $r^{-p}$, $p>d$, while the FM interaction has strength $J$. If $p>d+1$ and $J$ is large enough the ground state is FM, while if $d1$ the ground state has a series of transitions from an antiferromagnetic state of period 2 to $2h$--periodic states of blocks of sizes $h$ with alternating sign, the size $h$ growing when the FM interaction strength $J$ is increased (a generalization of this result to the case $0d$. The first term in (\ref{1.1}) will be called the {\it exchange energy} term and we will take $J\ge 0$ while the second one will be called the {\it dipolar energy} term. In $d>1$ we shall define $|i-j|$ to be the usual Euclidean distance between $i$ and $j$. Note that the sum over $n$ in the definition of $J_p(j-i)$ makes sense as long as $p>d$. However some of the results discussed below hold true even in the case $d-1< p\le d$, {\it provided} the definition of $J_p(j-i)$ is replaced by $J_p(j-i)=|j-i|^{-p}+(2N-|j-i|)^{-p}$, $\forall 1\le |j-i|\le 2N-1$. In the following we will restrict attention to the case $p>d$ and we will comment on possible generalizations of our results to $d-1d+1$ and $J$ large enough the ground state is ferromagnetic and is unique modulo a global spin flip. This is, in $d>1$, a corollary of the contour estimates in \cite{GGR66}, and we shall reproduce its proof as a byproduct of our analysis. For this case the low temperature Gibbs states are also known: for $d=1$ and $p>2$ there is a unique Gibbs state for any $\b<+\io$ \cite{R68} while for $d\ge 2$, $p>d+1$ and $\b$ large enough, there are two different pure states obtained as the thermodynamic limits of the equilibrium states with $+$ or $-$ boundary conditions \cite{GGR66}. On the contrary, for $dd$. For $d=1$ the characterization is complete, in the sense that for any $p>1$ we can compute the ground state energy per site and we can prove that the corresponding ground state configurations are periodic and consist of blocks of equal size and alternating signs (a block is a maximal sequence of adjacent spins of the same sign). \\ \\ {\bf Theorem 1.} (Ground state energy, d=1).\\ {\it For $d=1$ and $p>1$, the ground state energy $E_0(N)$ of the Hamiltonian (\ref{1.1}) satisfies % \begin{equation}\lim_{N\to\io}\frac1{2N} E_0(N)=\min_{h\in\zzz^+}e(h) \label{1.2a}\end{equation} % where $e(h)\defin\lim_{N\to\io} (2Nh)^{-1}E_{per}^{(h)}(Nh)$ and $E_{per}^{(h)}(Nh)$ is the energy of a periodic configuration on a ring of $2Nh$ sites consisting of blocks of size $h$ with alternating signs.} \\ The function $e(h)$ is explicitly computable, see Section \ref{sec3}. The minimization of $e(h)$ with respect to $h$ can be performed exactly and, as expected, for $p>2$ and $J$ larger than an explicit constant $J_0\=\G(p)^{-1}\int_0^\io d\a\a^{p-1}e^{-\a}(1-e^{-\a})^{-2}$, the minimizer is $h=+\io$ (\ie the ground state is ferromagnetic). For $12$ and $0\le J1$. At the values of $J$ such that the minimum of $e(h)$ is attained at a single integer $h^*(J)$, then in a ring of length $2N$, such that $N$ is divisible by $h^*(J)$, the only ground states are the periodic configurations consisting of blocks of size $h^*(J)$. At the values of $J$ such that the minimum is attained at two consecutive integers $h^*(J)$ and $h^*(J)+1$, then in a ring of length $2N$, such that $N$ is divisible both by $h^*(J)$ and $h^*(J)+1$, the only ground states are the periodic configurations consisting of blocks of the same size, either $h^*(J)$ or $h^*(J)+1$. In both cases there is a finite (independent of $N$) gap between the energies of the ground states and of any other state.}\\ \\ {\it Remarks.}\\ \01) {\it Stability for $01$ and, provided we change the definition of $J_p(j-i)$ as described after (\ref{1.1}), they can be easily adapted to cover the cases $00$ the ground state energy can be computed by minimizing energy among the periodic states of blocks with alternating signs and the computation shows that the ground state energy is {\it extensive} (\ie it scales proportionally to $N$ as $N\to \io$), even for $00$, the Hamiltonian (\ref{1.1}) is {\it not} reflection positive anymore. Still, one can think of making use of the reflection symmetry around bonds $b=(i,i+1)$ separating a spin $\s_i$ from a spin $\s_{i+1}=-\s_i$: such reflection does not change the exchange energy between $\s_i$ and $\s_{i+1}$ and lowers the dipole energy. By repeatedly reflecting around such sites one could expect to be able to reduce the search for the ground states just to the class of periodic configurations of blocks of the same size and alternating sign and explicitly look for periodic configuration with minimal energy (we recall that by block we mean a maximal sequence of consecutive spins all of the same sign). However there is a difficulty: because of periodic boundary conditions, in order not to increase the exchange energy in the reflection, one should reflect around bonds $b$ not only separating a $+$ from a $-$ spin, but with the further property that the bond $b'$ at a distance $N$ from $b$ also separates a $+$ from a $-$ spin: but in a generic configuration $\ss$ there will be no bond $b$ with such a property! A possible solution to this problem is to cut the ring into two uneven parts both containing the same number of blocks but not necessarily of the same length. The configurations obtained by reflecting the two uneven parts will have a lower energy but in general different lengths $2N',2N''$. The idea is to reflect repeatedly in this fashion, keeping track of the errors due to the fact that the length of the ring is changing at each reflection. A convenient way of doing this, exploited in Section \ref{sec3}, is to rewrite the spin Hamiltonian (\ref{1.1}) as an effective Hamiltonian for new ``atoms'' of ``charges'' $h_i$, corresponding to the spin blocks of size $h_i$. The new effective Hamiltonian $E(\ldots,h_{-1},h_0,h_1,h_2,\ldots)$ is again reflection symmetric (in a slightly different sense, though) and its explicit form allows for an easy control of the finite size errors one is left with after repeated reflections. Some technical aspects of the proofs of Theorem 1 and 2 are given in the Appendices. The proof of Theorem 3, in particular the lower bounds in (\ref{1.3aa}) and (\ref{1.3}), is based on a Peierls' contour estimate described in Section \ref{sec4}. For large ferromagnetic coupling $J$, we shall describe the ground state configuration(s) in terms of droplets of $-$ spins surrounded by $+$ spins and of contours separating the $+$ from the $-$ phases. The requirement that the energy of the ground state configuration is minimal imposes bounds on the geometry and the energy of such droplets, implying the lower bound in (\ref{1.3}). Contrary to the methods exploited in the proofs of Theorem 1 and 2, the proof of Theorem 3 is robust under modifications both of the boundary conditions and of the specific form of the interaction potential. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \*\section{One dimension}\lb{sec3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we want to prove Theorems 1 and 2. Restricting to $d=1$, we shall consider any configuration $\ss\in\{\pm1\}^{\L_N}$ on the ring of length $2N$ as a sequence of blocks of alternating sign, where by definition a block is a maximal sequence of adjacent spins of the same sign. We note that, due to the periodic boundary condition, the number $M$ of blocks on the ring is either $1$ (if the state is ferromagnetic) or an even number. We shall denote by $h_i$, $i=1,\ldots,M$, the sizes of such blocks. A block will be called a $+$ block ($-$ block), if its spins are all of sign $+1$ ($-1$). We want to prove that the sizes $h_i$ of the blocks in the ground state are all equal, at least for $N$ large enough. The strategy will be the following. Given any configuration $\ss_N$ in the ring $\L_N$ with $M\ge 2$ blocks of alternating signs of sizes $h_{-\fra{M}2+1},\ldots,h_{\fra{M}2}$, we will rewrite the energy $H_N(\ss_N)$ as a function of $M$ and the $h_i$'s, \ie $H_N(\ss_N)=E_N\big(M,(\ul h_L,\ul h_R)\big)$, where $\ul h_L=(h_{-\fra{M}2+1},\ldots,h_0)$ and $\ul h_R=(h_1,\ldots, h_{\fra{M}2})$. Setting $\ul h=(\ul h_L,\ul h_R)$, we will show that, for $M$ and $N$ fixed, the Hamiltonian $E_N(M,\ul h)$ is {\it reflection positive} with respect to the reflection % \begin{equation}\hat\th h_i=h_{-i+1}\;, \qquad -\fra{M}22$} Let us now discuss the case $p>2$. It is straightforward to verify that in this case, if $J\ge J_0\=\int_0^\io \frac{d\a}{\G(p)}\a^{p-1}\frac{e^{-\a}} {(1-e^{-\a})^2}$, then the minimizer of $e(h)$ is $h=+\io$. Correspondingly we can prove that the ground state of (\ref{1.1}) is the ferromagnetic state. In fact let us assume by contradiction that in the ground state there is a block $B_h$ of finite size $h$. Then it must be true that the energy $\D E(h)$ needed to reverse the sign of all the spins in $B_h$ is nonnegative: $\D E(h)\ge 0$. On the other side $\D E(h)\le -4J+2 E_1(h)$, where $E_1(h)$ is the dipole energy between the $+$ block $B_h$ and an external sea of $+$ spins. It is straightforward to check that % \begin{equation}E_1(h)= 2\int_0^\io\frac{d\a}{\G(p)}\a^{p-1}\frac{e^{-\a}} {(1-e^{-\a})^2}(1-e^{-\a h})<2J_0\label{3.22a}\end{equation} % and this leads to a contradiction. This proves Theorem 1 for $p>2$ and $J\ge J_0$. If on the contrary $J2$ and $J\ge J_0$ the statement is a corollary of the proof above: in fact the contradiction obtained after (\ref{3.22a}) shows that in this case the ferromagnetic state is the unique ground state. Let us then consider the cases $12$ and $J0$: this is proven in Appendix \ref{A5}. Then, using the fact that $|\III_0|+|\III_2|/2=M_0$ (with $M_0$ the number of blocks of size $h^*$) and that $|\III_1|+|\III_2|/2=M_1$ (with $M_1$ the number of blocks of size $h^*+1$), we get % \begin{equation}E_N(M,\ul h^0)-2N e_0(N)\ge |\III_2|(h^*+\fra12)(\wt e-\wt e_0) \label{3.30}\end{equation} % and the proof of Theorem 2 is concluded. \qed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \*\section{Higher dimensions}\lb{sec4} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we want to prove Theorem 3. As already remarked after Theorem 3, for $d=2$ and $p=3$ the upper bound was obtained in \cite{MWRD95} by minimizing over the periodic striped configurations (and checking that asymptotically for large $J$ such an upper bound is better than the one obtained by minimizing over the periodic checkerboard configurations). The upper bound in dimension higher than 2 and for $d \ell)J_p(i-j)\label{2.4} \end{equation} % Denoting the last term by $E^{>\ell}(\d)$, we note that $\sum_{\d\in\DD(\D)}E^{>\ell}(\d)$ can be bounded above by $2\min\{|\D|,|\D^c|\} \Phi_p(\ell)\le |\L|\Phi_p(\ell)$, where $\Phi_p(\ell)=\sum_{|n|>\ell}|n|^{-p}\le const.\ell^{-(p-d)}$. For large $\ell$ the constant is smaller than $2d|\SSS_d|/(p-d)$ where $|\SSS_d|$ is the volume of the $d$--dimensional unit sphere. The first term in the r.h.s. of (\ref{2.4}) can be rewritten as % \begin{equation}E^{\le \ell}(\d)= 2\sum_{|n|\le \ell}J_p(n)\sum_{i\in\d}\sum_{j\in\D^c} \openone(i-j=n)\le 2\sum_{|n|\le \ell}\frac1{|n|^p}\sum_{i\in\d}\sum_{j\in\zzz^d\setminus\d} \openone(i-j=n)\label{2.5}\end{equation} % where in the last inequality we neglected an error term vanishing in the thermodynamic limit. Now, the number of ways in which $n=(n_1,\ldots,n_d)$ may occur as the difference $i-j$ or $j-i$ with $i\in\d$ and $j\not\in\d$ is at most $\sum_{\g\in\G(\d)}\sum_{i=1}^d|\g|_i|n_i|$, where $|\g|_i$ is the number of faces in $\g$ orthogonal to the $i$--th coordinate direction. To prove this, draw a path on the lattice of length $|n_1|+\cdots+|n_d|$ connecting $i$ and $j$. Such a path must cross a face of $\g$ and if this face is orthogonal to the $i$--th coordinate axis, can do so only in $|n_i|$ ways and the claim follows. The conclusion is that % \begin{equation}E^{\le \ell}(\d) \le \sum_{\g\in\G(\d)}\sum_{|n|\le \ell}\fra1{|n|^p} \sum_{i=1}^2 |\g|_i|n_i|=\sum_{\g\in\G(\d)}|\g| \sum_{|n|\le \ell}\fra{|n_1|}{|n|^p}\;.\label{2.7}\end{equation} % The sum $\sum_{|n|\le \ell}|n_1|/|n|^p$ can be bounded above by $\int_{1\le|x|\le\ell}d^dx|x_1|/|x|^{p}+const.$ where the constant is smaller than $2pd|\SSS_d|2^{p-d}/(p-d)$ and $|\SSS_d|$ is the volume of the $d$--dimensional unit sphere. Then (\ref{2.7}) implies $E^{\le \ell}(\d)\le 2\Psi_p(\ell)+2pd|\SSS_d|2^{p-d}/(p-d)$, where $\Psi_p(\ell)=|\SSS_{d-1}|\log\ell$ if $p=d+1$ and $\Psi_p(\ell)=|\SSS_{d-1}|(\ell^{d+1-p}-1)/(d+1-p)$ if $dd+1$ and $J$ is large enough no droplet can appear in the ground state, \ie the ground state is ferromagnetic. In fact, let us assume by contradiction that the ground state is not $\s_i\=-1$ and that there is at least one droplet $\d$ in the ground state. Then the energy needed to reverse all the spins in $\d$ must be positive: $-2J\sum_{\g\in\G(\d)}|\g|+ E_{dip}(\d)\ge0$. Proceeding as above in the proof of (\ref{2.7}) and using that $p>d+1$, we get $E_{dip}(\d)\le \sum_{\g\in\G(\d)}|\g| \sum_{|n|\ge 1}\fra{|n_1|}{|n|^p}\le const. \sum_{\g\in\G(\d)}|\g|$; hence for $J$ big enough we get a contradiction and this proves that the ground state is ferromagnetic.\\ The contour method implemented in the proof of Theorem 3 also allows to get some informations about the geometry of contours and droplets in the ground state for $d1$, with alternating sign, the size $h$ changing (increasing discontinuously) when the ferromagnetic interaction strength $J$ is increased. Our proof is based on a reflection positivity argument which relies heavily on the details of the model Hamiltonian, \eg on the fact that the long--range repulsion is reflection positive. It is an interesting open problem to establish more general conditions the long range repulsive interaction should satisfy in order to guarantee existence of a periodic modulated ground state. Note that the problem of determining the ground state of a spin system with positive and convex potential was solved by Hubbard and by Pokrovsky and Uimin (even in presence of a magnetic field) \cite{Hu78,PU78}: this means that for $J=0$ the assumption of convexity of the potential is enough to determine the ground state of (\ref{1.1}). The proof in \cite{Hu78,PU78} was generalized to an ``almost'' convex case by Jedrzejewski and Miekisz \cite{JM001,JM002}. However this proof requires a small ferromagnetic interaction $1\le J\le 1+2^{-p}$. It is an open problem to generalize our analysis (in the presence of a large ferromagnetic coupling) to the case of a long range convex interaction. It would also be interesting to establish what would be the effect of adding a magnetic field to the model Hamiltonian. It is expected that in the presence of a large short range ferromagnetic coupling and of a positive magnetic field $B$ of increasing strength the ground state is still periodic with the $+$ blocks larger than the $-$'s, at least if $|B|$ is not too large \cite{GD82}. However for large magnetic field it could be the case that small variations of the magnetic field could induce an infinite sequence of transitions between periodic states characterized by different rational values of the magnetization (Devil's staircase): this is in fact what happens for $J=0$ as a function of the magnetic field \cite{BB82}. According to \cite{BB82} the antiferromagnetic state would remain the ground state for $|B|1$ there is a unique limit Gibbs state at all values of the inverse temperature $\b$ \cite{FVV82,BS83,K99}. The proof in \cite{K99} does not extend to the case $J>0$. It is known however that for $J>0$ all Gibbs states are translation invariant \cite{FVV82}. Similarly, using the argument in \cite{BLP79}, one can show that the Gibbs states obtained as limits of finite volume Gibbs measures with boundary conditions $\t^k\ul\s_{per}(h^*(J))$, where $\ul\s_{per}(h^*(J))$ is a periodic ground state configuration with blocks of size $h^*(J)$ and $\t$ the translation operator, are all equivalent among each other, for any $k=1,\ldots,h^*(J)$. If, on the contrary, $0i$, we can simply apply the definition and (\ref{A4.1}) to find: % \bea&&\e_i\e_j\sum_{n\in \zzz}\sum_{l=1}^{h_i}\sum_{m=1}^{h_j} \fra1{|m-l+h_i+\cdots+h_{j-1}+2nN|^p}=\label{A4.2}\\ %&=(-1)^{j-1}\sum_{l=1}^{h_i}\sum_{m=1}^{h_j}\sum_{n\ge 0} %\fra1{(m-l+h_i+\cdots+h_{j-1}+2nN)^p}+\cr %&+(-1)^{j-1}\sum_{m=1}^{h_j}\sum_{l=1}^{h_i}\sum_{n\le -1} %\fra1{(l-m+h_j-h_i-\cdots-h_{j}-2(n+1)N)^p}=\cr &&=\e_i\e_j\int_0^\io\fra{d\a}{\G(p)}\,\a^{p-1} \fra{e^{-\a}}{(1-e^{-\a})^2}(1-e^{-\a h_i}) (1-e^{-\a h_j})\Big[\prod_{i0$. Using the general expression (\ref{3.11}), after some algebra we find that % \bea \wt e&&=-J+A+\fra{4J}{2h^*+1}-\fra2{2h^*+1}\int_0^\io d\a\,\n(\a) \Biggl\{\fra{2(1-e^{-\a h^*})(1-e^{-\a (h^*+1)})} {1-e^{-\a(4h^*+2)}}+\nn\\ &&+\fra{e^{-2\a h^*}(1-e^{-\a (h^*+1)})^2+e^{-2\a(h^*+1)}(1-e^{-\a h^*})^2} {1-e^{-\a(4h^*+2)}}\Biggr\}\label{A5.1}\eea % where $\n(\a)\=(\G(p))^{-1}\a^{p-1} e^{-\a}(1-e^{-\a})^{-2}$. We want to prove that this expression is strictly larger than $\wt e_0$. Note that, since both $h^*$ and $h^*+1$ are minimizers, we have % \begin{equation}\fra{J}{h^*}-\fra1{h^*}\int_0^\io d\a\, \n(\a)\tanh\fra{\a h^*}2= \fra{J}{h^*+1}-\fra1{h^*+1}\int_0^\io d\a\,\n(\a)\tanh\fra{\a (h^*+1)}2 \label{A5.2}\end{equation} % implying that % \begin{equation}J=\int_0^\io d\a\,\n(\a)\Big[(h^*+1)\tanh\fra{\a h^*}2-h^* \tanh\fra{\a (h^*+1)}2\Big]\label{A5.3}\end{equation} % and % \begin{equation} \wt e_0=2\int_0^\io d\a\, \n(\a)\Big(\tanh\fra{\a h^*}2-\tanh\fra{\a (h^*+1)}2\Big) \label{A5.4}\end{equation} % Using (\ref{A5.3}) and (\ref{A5.4}) we find % \bea&&\wt e-\wt e_0=2\int_0^\io d\a\,\n(\a) \Bigg[ \tanh\fra{\a h^*}2+\tanh\fra{\a (h^*+1)}2-\lb{A5.5}\\ &&-\fra{2(1-e^{-\a h^*})(1-e^{-\a (h^*+1)})+ e^{-2\a h^*}(1-e^{-\a (h^*+1)})^2+e^{-2\a(h^*+1)}(1-e^{-\a h^*})^2} {1-e^{-\a(4h^*+2)}}\Bigg]\nn\eea % A bit more of algebra shows that (\ref{A5.5}) can be rewritten as % \begin{equation}\wt e-\wt e_0=2\int_0^\io d\a\,\n(\a) (e^{-\a h^*}-e^{-\a(h^*+1)})^2 (1-e^{-\a h^*})(1-e^{-\a (h^*+1)})\label{A5.6}\end{equation} % and this concludes the proof. \acknowledgments We would like to thank G. Gallavotti, T. Kuna, K. Khanin and A.C.D. van Enter for useful discussions and comments. The work of JLL was supported by NSF Grant DMR-044-2066 and by AFOSR Grant AF-FA 9550-04-4-22910 and was completed during a visit to the IAS. The work of AG and EHL was partially supported by U.S. National Science Foundation grant PHY 01 39984, which is gratefully acknowledged. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation. \bibliography{ising} \end{document} ---------------0604281027472--